Asymptotic series

Stream: maths

[Yury G. Kudryashov]

I'm going to formalize asymptotic series of a function along a filter, with lemmas like uniqueness and "if there exists an asymptotic series up to each finite order, then they all glue together into 1 asymptotic series". Is there any standard reference on this topic, or should I start with some definition that fits my use cases and, if needed, generalize later?

[Yury G. Kudryashov]

Use cases that I care about:

• Taylor series;
• "Dulac series" $f(z)=az+b+\sum_k P_k(z)e^{-\nu_k z}$, where $0<\nu_0<\nu_1<\dots$, in a domain $C(1+\sqrt{|\Im z|})\le \Re z$ (appears in Ilyashenko's proof ot the fact that an analytic vector field on the sphere with hyperbolic singular pts only can't have infinitely many limit cycles).

[Michael Stoll]

I think analytic number theorists also care about series involving powers of log (or even log log). E.g. like this.

[Sebastien Gouezel]

Eberl uses a very general notion of asymptotic series in http://cl-informatik.uibk.ac.at/users/meberl//pdfs/real_asymp.pdf

[Yury G. Kudryashov]

@Michael Stoll The series above is in fact in the logarithmic chart, so in the original chart it would use polynomials of logarithms multiplied by real powers.